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A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in each degree. My question is: does the converse hold? that is, is every simplicial set, having a finite number of simplicies in each degree, necessarily finite?

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Just a remark: it is not true that every simplicial set is the colimit of its non-degenerate simplices. For example, let $X$ be obtained from the standard $2$-simplex by collapsing the edge connecting $0$ and $1$. The colimit of non-degenerate simplices of $X$ is the standard $2$-simplex with vertices $0$ and $1$ identified, but the edge connecting them is not collapsed this time. – Karol Szumiło Nov 17 '13 at 17:08
Further to what Karol says, the construction sending a simplicial set $X$ to the colimit of its non-degenerate simplices is not even fully functorial in $X$, if I’m not mistaken. It only respects maps that preserve non-degenerate simplices. – Peter LeFanu Lumsdaine Nov 17 '13 at 17:16
In Lurie's book "higher topos theory" Variant it is mentioned that if $K$ is a finite simplicial set such that every nondegenerate simplex $\Delta^n\to K$ is a monomorphism, then the full subcategory inclusion of the non degenerate simplices of $K$, in the hole simplex category of $K$, is cofinal. I forgot this extra "monomotphism" condition, so thanks. Anyway, the fact that every finite simplicial set has only a finite number of simplicies in each degree is true. – Ilan Barnea Nov 17 '13 at 20:09
Hi Ilan, Have you tried to construct an infinite simplicial set with finitely many simplicies in each degree? I think you will find that you can in fact do so (e.g. $\mathbb{CP}^\infty$, and more generally classifying spaces of finite groups). But if you failed to find any, perhaps you can study where the obstructions come from? As is stands, the question looks like you haven't put much thought into finding the answer. – Theo Johnson-Freyd Nov 17 '13 at 23:22
up vote 7 down vote accepted

Let G be a finite group viewed as a one object category. Then the nerve BG is a simplicial set with finitely many simplices in each dimension but it is not finite.

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Take $X$ to be the “infinite-dimensional dunce’s cap”, with a unique non-degenerate simplex $x_n$ in each dimension, and with every face of $x_n$ equal to $x_{n-1}$.

Explicitly, $X_n = \coprod_{m \leq n} \mathrm{Surj}([n],[m])$. So it’s clear that this has finitely many simplices in each dimension, but infinitely many non-degenerate ones in total.

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Another example in the same spirit is an infinite wedge $\bigvee S^n$ of spheres of different dimension. – Eric Wofsey Nov 17 '13 at 18:11
Simpler: $\Delta[n]\subset \Delta[n+1]$ and then consider the colimit of the tower. – Philippe Gaucher Nov 18 '13 at 8:56
@PhilippeGaucher: that was my first thought too, but it doesn’t work — it ends up having infinitely many n-d simplices in each dimension. – Peter LeFanu Lumsdaine Nov 18 '13 at 14:36
@PeterLeFanuLumsdaine yes indeed. – Philippe Gaucher Nov 18 '13 at 19:03

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